Introduction to QM/MM Simulations

Chapter 3 Introduction to QM/MM Simulations Gerrit Groenhof Abstract Hybrid quantum mechanics/molecular mechanics (QM/MM) simulations have become a popular tool for investigating chemical reactions in condensed phases. In QM/MM methods, the region of the system in which the chemical process takes place is treated at an appropriate level of quantum chemistry theory, while the remainder is described by a molecular mechanics force field. Within this approach, chemical reactivity can be studied in large systems, such as enzymes. In the first part of this contribution, the basic methodol- ogy is briefly reviewed. The two most common approaches for partitioning the two subsystems are presented, followed by a discussion on the different ways of treating interactions between the subsystems. Special attention is given on how to deal with situations in which the boundary between the QM and MM subsystems runs through one or more chemical bonds. The second part of this contribution discusses what properties of larger system can be obtained within the QM/MM framework and how. Finally, as an example of a QM/MM application in practice, the third part presents an overview of recent QM/MM molecular dynamics simulations on photobiological systems. In addition to providing quantities that are experimentally accessible, such as structural intermediates, fluorescence lifetimes, quantum yields and spectra, the QM/MM simulations also provide information that is much more difficult to measure experimentally, such as reaction mechanisms and the influence of individual amino acid residues. Key word: Quantum mechanics, Molecular mechanics, QM/MM, Molecular dynamics 1. Introduction In this chapter we present a short introduction into the development and application of computational techniques for modelling chemical reactions in the condensed phase. We start by reviewing the basic concepts of these methods. We then discuss how these methods can be used in practical computations and what kind of information can be obtained. We conclude this chapter with a short review of an application on a photobiological system, for which the simulations not only revealed the detailed sequence of events that follow photon absorption but also demonstrate how the biological environment controls the photochemical reaction. Luca Monticelli and Emppu Salonen (eds.), Biomolecular Simulations: Methods and Protocols, Methods in Molecular Biology, vol. 924, DOI 10.1007/978-1-62703-017-5_3, # Springer Science+Business Media New York 2013 43 44 G. Groenhof 2. QM/MM: Theory and Implementation The size and complexity of a typical biomolecular system, together with the timescales that must be reached, necessitate the use of classical molecular dynamics for the nuclear degrees of freedom. In molecular dynamics (MD) simulations, Newton’s equations of motion are solved numerically to obtain a trajectory of the dynamics of a molecule over a period of time (1). To model electronic rearrangements during a chemical reaction, a quantum mechanical description (QM) is required for those parts of the system that are involved in the reaction. For the remainder, a simple molecular mechanics force field model suffices (MM). The interactions in the system are thus computed within a hybrid QM/MM framework. 2.1. Molecular Molecular dynamics simulations of biological systems have come of Mechanics age (2). Since the first application of MD on a small protein in vacuum more than three decades ago (3), advances in computer power, algorithmic developments and improvements in the accu- racy of the used interaction functions have established MD as an important and predictive technique to study dynamic processes at atomic resolution (4). In the interaction functions, the so-called molecular mechanics force field, simple chemical concepts are used to describe the potential energy of the system (1): X X X ¼ N bonds bond þ N angles þ N torsions torsion V MM V i V j V l X i X j X X l N N N N þ MM MM V Coul þ MM MM V LJ; (1) i j>i ij i j>i ij where NMM is the number of atoms in the system. Bonds and angles (V bond, Vangle) are normally modelled by harmonic functions, and torsions by periodic functions (V torsion). The pairwise electrostatic interaction between atoms with a partial charge (Qi) is given by Coulomb’s law: e2Q Q Coul ¼ i j V ij (2) 4pE0Rij ; in which Rij denotes the interatomic distance, e the unit charge and E0 the dielectric constant. Van der Waals interactions, for example the combination of short-range Pauli repulsion and long-range dispersion attraction, are most often modelled by the Lennard- Jones potential: ! ! ij 12 ij 6 LJ ¼ C12 À C6 ; V ij (3) Rij Rij 3 Introduction to QM/MM Simulations 45 ij ij with C12 and C6 a repulsion and attraction parameter, respectively, which depend on the atomtypes of the atoms i and j. Electrons are thus ignored in molecular mechanics force fields. Their influence is expressed by empirical parameters that are valid for the ground state of a given covalent structure. Therefore, processes that involve electronic rearrangements, such as chemical reactions, cannot be described at the MM level. Instead, these processes require a quantum mechanics description of the electronic degrees of freedom. However, the computational demand for evaluating the electronic structure places severe constraints on the size of the system that can be studied. 2.2. Hybrid Quantum Most biochemical systems, such as enzymes, are too large to be Mechanics/Molecular described at any level of ab initio theory. At the same time, the Mechanics Models available molecular mechanics force fields are not sufficiently flexible to model processes in which chemical bonds are broken or formed. To overcome the limitations of a full quantum mechanical description on the one hand, and a full molecular mechanics treatment on the other hand, methods have been developed that treat a small part of the system at the level of quantum chemistry (QM), while retaining the computationally cheaper force field (MM) for the larger part. This hybrid QM/MM strategy was originally introduced by Warshel and Levitt (5) and is illustrated in Fig. 1. The justification for dividing a system into regions that are described at different levels of theory is the local character of most chemical reactions in condensed phases. A distinction can usually be made between a “reaction centre” with atoms that are directly involved in the reaction and a “spectator” region, in which the atoms do not directly participate in the reaction. For example, most reactions in solution involve the reactants and the first few solvation shells. The bulk solvent is hardly affected by the reaction, but can influence the reaction via long-range interactions. The same is true for most enzymes, in which the catalytic process is restricted to an active site located somewhere inside the protein. The rest of the protein provides an electrostatic background that may or may not facilitate the reaction. Fig. 1. Illustration of the QM/MM concept. A small region, in which a chemical reaction occurs and therefore cannot be described with a force field, is treated at a sufficiently high level of QM theory. The remainder of the system is modelled at the MM level. 46 G. Groenhof The hybrid QM/MM potential energy contains three classes of interactions: interactions between atoms in the QM region, between atoms in the MM region and interactions between QM and MM atoms. The interactions within the QM and MM regions are relatively straightforward to describe, that is at the QM and MM level, respectively. The interactions between the two subsystems are more difficult to describe, and several approaches have been proposed. These approaches can be roughly divided into two categories: subtractive and additive coupling schemes. 2.3. Subtractive In the subtractive scheme, the QM/MM energy of the system is QM/MM Coupling obtained in three steps. First, the energy of the total system, con- sisting of both QM and MM regions, is evaluated at the MM level. The QM energy of the isolated QM subsystem is added in the second step. Third, the MM energy of the QM subsystem is computed and subtracted. The last step corrects for including the interactions within the QM subsystem twice: V QM=MM ¼ V MMðMM þ QMÞþV QMðQMÞV MMðQMÞ: (4) The terms QM and MM stand for the atoms in the QM and MM subsystems, respectively. The subscripts indicate the level of theory at which the potential energies (V ) are computed. The most widely used subtractive QM/MM scheme is the ONIOM method, developed by the Morokuma group (6, 7), and is illustrated in Fig. 2. The main advantage of the subtractive QM/MM coupling scheme is that no communication is required between the quantum chemistry and molecular mechanics routines. This makes the implementation relatively straightforward. However, compared to the more advanced schemes that are discussed below, there are also disadvantages. A major disadvantage is that a force field is required for the QM subsystem, which may not always be available. In addition, the force field needs to be sufficiently flexible to describe the effect of chemical changes when a reaction occurs. δ δ + δ+ δ– MM + δ+ MM δ+ δ– δ– δ+ δ– δ+ δ+ δ+ δ+ δ δ– – δ– δ+ δ– δ+ = + δ+ δ+ δ+ δ+ – δ– δ– δ δ δ– + δ– + δ+ δ+ QM δ– QM δ– MM δ– δ– δ δ– δ δ– + δ+ + δ+ Fig. 2. Subtractive QM/MM coupling: The QM/MM energy of the total system (left hand side of the equation) is assumed to be equal to the energy of the isolated QM subsystem, evaluated at the QM level, plus the energy of the complete system evaluated at the MM level, minus the energy of the isolated QM subsystem, evaluated at the MM level.

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